When the temperature of a solid material is increased the material expands resulting in an increase in volume. When the temperature of a solid material is decreased the material contracts resulting in a decrease in volume. It is a well know fact that solid materials typically expand in response to heating and contract upon cooling. Materials expand because an increase in temperature results in greater thermal vibration of the atoms in the material and subsequently an increase in the average separation distance between adjacent atoms. The dimensional response of a solid material due to a change in temperature is characterized by the coefficient of thermal expansion (CTE). The linear CTE describes the magnitude of expansion as a function of increased temperature and is expressed as follows:
  α  =                    Δ        ⁢                                  ⁢        l                    l        ⁢                                  ⁢        Δ        ⁢                                  ⁢        T              .  where                α=CTE        Δl=change in length        l=initial length, and        ΔT=change in temperature.For an area expansion in which the area, A=l2        
            Δ      ⁢                          ⁢      A        A    =      2    ⁢    α    ⁢                  ⁢    Δ    ⁢                  ⁢    T  where                ΔA=change in area        A=initial area,        ΔT=change in temperature, and        α=CTE.For a volume expansion        
            Δ      ⁢                          ⁢      V        V    =      3    ⁢    αΔ    ⁢                  ⁢    T  where                ΔV=change in volume        V=initial volume        ΔT=change in temperature, and        α=CTE.The magnitude of the CTE depends on the atomic structure and bonding of the material. A weakly bonded solid will have a higher CTE than a strongly bonded solid. Different materials have different binding forces and thus different rates of expansion upon heating. For example, strongly bonded insulators, such as ceramics, have relatively low CTEs compared to metals.        
It is a common practice in the development or design of a part or system to pick a material with the best properties for the given application. These properties can be physical such as strength and density or economical such as cost and availability. Differing materials are often used in one mechanical system to optimize different parts of the system. In these systems different material parts will be fastened together. These differing materials will have differing thermal expansions. When this system experiences a change in temperature the differing materials will have differing amounts of thermal strain or thermally induced change in length. Since the materials are rigidly fastened together this strain becomes a stress in the parts based on Hooke's law,σ=E(ε1−ε2)where                σ=stress        E=modulus of elasticity        ε1=strain material 1        ε2=strain material 2.        
Most mechanical systems only require that the fastened joint does not fail under the thermal stresses. This is easily accomplished by altering the shape and size of the fastened components and the fasteners. However, for stress sensitive instruments, very small stresses in the components can cause instrument errors and failure.
Current methods to compensate for the stress and deformation of joined structures generally address linear expansion and contraction. For example, steel railroad rails, if laid in contact end to end would buckle on hot days as the result of thermal expansion. Most tracks are built from pieces of steel supported by wooden ties, and laid with a space between the ends. This space provides a buffer for thermal expansion, allowing the rail to elongate without contacting the next rail. Another example of a method to compensation for thermal expansion is expansion joints. Bridges are built with metal expansion joints, which contain gaps between bridge sections that allow for expansion and contraction without causing faults in the overall structure of the bridge.
For cryogenic optical testing chambers in which high performance Silicon Carbide mirrors are used, it is desired to reduce mirror errors caused by thermal gradients. However, the chamber is built primarily of aluminum and stainless steel which have CTEs that are much different than that of the Silicon Carbide mirrors. Mounting the Silicon Carbide mirrors to the chamber and expecting good performance over a 250 K range is difficult to achieve.
When materials with different CTE are joined, stress and deformation occurs as the result of changes in temperature. Systems have been designed to incorporate a material with an intermediate CTE to reduce the magnitude of the thermal stresses. For example, struts can be designed with a controlled thermal expansion structure that maintains a constant length over a temperature range. If the surfaces of two materials with different CTEs need to be joined, one or more different materials with intermediate CTEs are usually sandwiched between the two surfaces to distribute the thermal stresses at the interfaces, thus resulting is a lower overall stress. Other available technologies to fasten materials with different CTEs involve using flexures to absorb the thermal elastic strain associated with differential thermal expansion rates.
Deficiencies with current methods and technologies for addressing thermal induced stresses in joined structures are associated with the linear nature of the solutions which are not applicable to non-linear structures. For non-linear structures, the use of an intermediate layer distributes the thermal induced stress to the interfaces but only reduces the magnitude of the thermal stress in the structure. Therefore a thermal expansion compensation method and system is needed to compensate for, and eliminate, thermal induced stresses in systems composed of materials with different coefficients of thermal expansion.